<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 5.4.2">
  <link rel="apple-touch-icon" sizes="180x180" href="/blog/images/icon.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/blog/images/icon.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/blog/images/icon.png">
  <link rel="mask-icon" href="/blog/images/icon.svg" color="#222">

<link rel="stylesheet" href="/blog/css/main.css">


<link rel="stylesheet" href="/blog/lib/font-awesome/css/all.min.css">

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"bgape002.gitee.io","root":"/blog/","scheme":"Gemini","version":"7.8.0","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":true,"show_result":true,"style":"mac"},"back2top":{"enable":true,"sidebar":false,"scrollpercent":true},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},"path":"search.xml"};
  </script>

  <meta name="description" content="1. 基本概率论公式1.加减法公式    加法公式$$P(A+B)&#x3D;p(A)+P(B)-P(AB) \tag{0.0}$$ ​    减法公式$$P(A-B)&#x3D;p(A)-P(AB) \tag{0.1}$$2. 条件概率    事件A已经发生的条件下，事件B发生的概率：$$P(B|A) &#x3D; \frac{P(AB)}{P(A)} \tag{1}$$">
<meta property="og:type" content="article">
<meta property="og:title" content="概率论">
<meta property="og:url" content="https://bgape002.gitee.io/2020/11/09/%E6%A6%82%E7%8E%87%E8%AE%BA/index.html">
<meta property="og:site_name" content="bgape002">
<meta property="og:description" content="1. 基本概率论公式1.加减法公式    加法公式$$P(A+B)&#x3D;p(A)+P(B)-P(AB) \tag{0.0}$$ ​    减法公式$$P(A-B)&#x3D;p(A)-P(AB) \tag{0.1}$$2. 条件概率    事件A已经发生的条件下，事件B发生的概率：$$P(B|A) &#x3D; \frac{P(AB)}{P(A)} \tag{1}$$">
<meta property="og:locale" content="zh_CN">
<meta property="article:published_time" content="2020-11-09T08:55:31.000Z">
<meta property="article:modified_time" content="2022-01-27T02:32:14.780Z">
<meta property="article:author" content="bgape002">
<meta property="article:tag" content="概率论">
<meta name="twitter:card" content="summary">

<link rel="canonical" href="https://bgape002.gitee.io/2020/11/09/%E6%A6%82%E7%8E%87%E8%AE%BA/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'zh-CN'
  };
</script>

  <title>概率论 | bgape002</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/blog/" class="brand" rel="start">
      <span class="logo-line-before"><i></i></span>
      <h1 class="site-title">bgape002</h1>
      <span class="logo-line-after"><i></i></span>
    </a>
      <p class="site-subtitle" itemprop="description">淡泊明志，宁静致远</p>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>




<nav class="site-nav">
  <ul id="menu" class="main-menu menu">
        <li class="menu-item menu-item-home">

    <a href="/blog/" rel="section"><i class="fa fa-home fa-fw"></i>首页</a>

  </li>
        <li class="menu-item menu-item-tags">

    <a href="/blog/tags/" rel="section"><i class="fa fa-tags fa-fw"></i>标签<span class="badge">66</span></a>

  </li>
        <li class="menu-item menu-item-categories">

    <a href="/blog/categories/" rel="section"><i class="fa fa-th fa-fw"></i>分类<span class="badge">27</span></a>

  </li>
        <li class="menu-item menu-item-archives">

    <a href="/blog/archives/" rel="section"><i class="fa fa-archive fa-fw"></i>归档<span class="badge">61</span></a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>
</nav>



  <div class="search-pop-overlay">
    <div class="popup search-popup">
        <div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container">
    <input autocomplete="off" autocapitalize="off"
           placeholder="搜索..." spellcheck="false"
           type="search" class="search-input">
  </div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div id="search-result">
  <div id="no-result">
    <i class="fa fa-spinner fa-pulse fa-5x fa-fw"></i>
  </div>
</div>

    </div>
  </div>

</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content post posts-expand">
            

    
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="https://bgape002.gitee.io/2020/11/09/%E6%A6%82%E7%8E%87%E8%AE%BA/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/blog/images/head.png">
      <meta itemprop="name" content="bgape002">
      <meta itemprop="description" content="mail: bgape002@163.com">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="bgape002">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          概率论
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-calendar"></i>
              </span>
              <span class="post-meta-item-text">发表于</span>

              <time title="创建时间：2020-11-09 16:55:31" itemprop="dateCreated datePublished" datetime="2020-11-09T16:55:31+08:00">2020-11-09</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="far fa-calendar-check"></i>
                </span>
                <span class="post-meta-item-text">更新于</span>
                <time title="修改时间：2022-01-27 10:32:14" itemprop="dateModified" datetime="2022-01-27T10:32:14+08:00">2022-01-27</time>
              </span>
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-folder"></i>
              </span>
              <span class="post-meta-item-text">分类于</span>
                <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
                  <a href="/blog/categories/math/" itemprop="url" rel="index"><span itemprop="name">math</span></a>
                </span>
            </span>

          
            <span class="post-meta-item" title="阅读次数" id="busuanzi_container_page_pv" style="display: none;">
              <span class="post-meta-item-icon">
                <i class="fa fa-eye"></i>
              </span>
              <span class="post-meta-item-text">阅读次数：</span>
              <span id="busuanzi_value_page_pv"></span>
            </span>
  
  <span class="post-meta-item">
    
      <span class="post-meta-item-icon">
        <i class="far fa-comment"></i>
      </span>
      <span class="post-meta-item-text">Valine：</span>
    
    <a title="valine" href="/blog/2020/11/09/%E6%A6%82%E7%8E%87%E8%AE%BA/#valine-comments" itemprop="discussionUrl">
      <span class="post-comments-count valine-comment-count" data-xid="/blog/2020/11/09/%E6%A6%82%E7%8E%87%E8%AE%BA/" itemprop="commentCount"></span>
    </a>
  </span>
  
  

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <h3 id="1-基本概率论公式"><a href="#1-基本概率论公式" class="headerlink" title="1. 基本概率论公式"></a>1. 基本概率论公式</h3><p>1.加减法公式<br>    加法公式<br>$$<br>P(A+B)=p(A)+P(B)-P(AB) \tag{0.0}<br>$$</p>
<p>​    减法公式<br>$$<br>P(A-B)=p(A)-P(AB) \tag{0.1}<br>$$<br>2. 条件概率<br>    事件A已经发生的条件下，事件B发生的概率：<br>$$<br>P(B|A) = \frac{P(AB)}{P(A)} \tag{1}<br>$$</p>
<span id="more"></span>
<ol start="3">
<li><p>乘法定理<br>$$<br>P(AB)=P(B|A)P(A) \tag{2}<br>$$</p>
</li>
<li><p>全概率公式</p>
<p>试验E的<strong>样本空间S</strong>，事件$B_i,i=1,2…,n$是样本空间的一个<strong>划分</strong>，每次试验有且仅有一个发生。</p>
<ul>
<li><p>$B_iB_j=\emptyset,i\neq j$</p>
</li>
<li><p>$B_1\bigcup B_2\bigcup…\bigcup B_n=S$</p>
</li>
</ul>
<p> 如果A是E的事件，事件A发生<br>$$<br>P(A)=P(A|B_1)P(B_1)+P(A|B_2)P(B_2)+…+P(A|B_n)P(B_n) \tag{3}<br>$$</p>
</li>
<li><p>贝叶斯公式<br>$$<br>P(B_i|A)=\frac{P(A|B_i)P(B_i)}{\Sigma_{j=1}^n P(A|B)P(B_j)+P(A|B_j)P(B_j)} \tag{4.0}<br>$$<br>​    n=2时，<br>$$<br>P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B)+P(A|\overline{B})P(\overline{B})} \tag{4.1}<br>$$</p>
</li>
</ol>
<h3 id="1-2-期望、方差、协方差、相关系数"><a href="#1-2-期望、方差、协方差、相关系数" class="headerlink" title="1.2 期望、方差、协方差、相关系数"></a>1.2 期望、方差、协方差、相关系数</h3><h4 id="期望"><a href="#期望" class="headerlink" title="期望"></a>期望</h4><p>在概率论和统计学中，数学期望（或均值）是试验中每次可能结果的概率乘以其结果的总和，是最基本的数学特征之一。<strong>它反映随机变量平均取值的大小</strong>。</p>
<ol>
<li>离散型随机变量X的数学期望：</li>
</ol>
<p>$$<br>E(X)=\sum^{\infty}_{k=1}x_kp_k<br>$$</p>
<ol start="2">
<li>连续型随机变量X的数学期望：<br>$$<br>E(x)=\int_{-\infty}^{+\infty}xf(x)dx<br>$$</li>
<li>期望的性质<br>   $$<br>   \begin{align}<br>   &amp;0. \qquad E(C) = C，C是常数 \<br>   &amp;1. \qquad E(CX) = CE(X) \<br>   &amp;2. \qquad E(X+Y) = E(X)+E(Y) \<br>   &amp;3. \qquad E(XY) = E(X)E(Y),X,Y独立<br>   \end{align}<br>   $$</li>
</ol>
<h4 id="方差"><a href="#方差" class="headerlink" title="方差"></a>方差</h4><p>方差是各变量值与其均值离差平方的平均数，它是测算数值型数据离散程度的最重要的方法。记为：<br>$$<br>\begin{align}<br>D(X)&amp;=E[X-E(X)]^2 \<br>&amp;=E(X^2)-[E(X)]^2<br>\end{align}<br>$$<br>标准差为方差的算术平方根。由于方差是数据的平方，与检测值本身相差太大，人们难以直观地衡量，所以常用均方差代替方差判断数据的波动。<br>$$<br>\sigma(X)=\sqrt{E[X-E(X)]^2}<br>$$<br>方差的性质：<br>$$<br>\begin{align}<br>&amp;0. \qquad D(C)=C,C是常数\<br>&amp;1. \qquad \cases{D(CX)=C^2D(X) \ D(X+C)=D(X)}\<br>&amp;2. \qquad \cases{D(X+Y)=D(X)+D(Y)+2Cov\ D(X+Y)=D(X)+D(Y),X,Y独立}\<br>&amp;3. \qquad D(X) \Leftrightarrow P{X=E(X)}=1<br>\end{align}<br>$$</p>
<h4 id="协方差"><a href="#协方差" class="headerlink" title="协方差"></a>协方差</h4><p>在概率论和统计学中，协方差用于衡量两个变量的总体误差。<br>$$<br>\begin{align}<br>Cov(X,Y)&amp;=E[(X-E(X))\cdot (Y-E(Y))] \<br>&amp;=E(XY)-E(X)(Y)<br>\end{align}<br>$$<br>当X=Y时：<br>$$<br>Cov(X,Y)=E[(X-E(X))]^2=E[(Y-E(Y))]^2=D(X)=D(Y)<br>$$</p>
<h4 id="相关系数"><a href="#相关系数" class="headerlink" title="相关系数"></a>相关系数</h4><p>相关系数是用以反映变量之间相关关系密切程度的统计指标。<br>$$<br>\rho_{XY}=\frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}<br>$$</p>
<p>相关系数性质：<br>$$<br>\begin{align}<br>&amp;0. \qquad |\rho_{XY}|\leqslant 1,即\rho_{XY} \in [-1,1] \<br>&amp;1. \qquad \cases{\rho_{XY}=0\Leftrightarrow X,Y不相关 \ |\rho_{XY} =1| \Leftrightarrow X,Y完全相关}\<br>&amp;2. \qquad 若Y=a+bX,则有 \<br>&amp; \qquad\qquad \cases{1,b&gt;0\0,b=0\-1,b&lt;0}\<br>\end{align}<br>$$</p>
<h4 id="协方差矩阵…"><a href="#协方差矩阵…" class="headerlink" title="协方差矩阵…"></a>协方差矩阵…</h4><p>计算各维度间的相关性。</p>
<h3 id="2-概率分布"><a href="#2-概率分布" class="headerlink" title="2. 概率分布"></a>2. 概率分布</h3><h4 id="2-1-两点分布-伯努利分布"><a href="#2-1-两点分布-伯努利分布" class="headerlink" title="2.1 两点分布(伯努利分布)"></a>2.1 两点分布(伯努利分布)</h4><p>随机变量X只取0和1两个值，X的概率函数为：<br>$$<br>P(x)=\cases{p^xq^{1-x},x=0,1\<br>0,x\neq 0,1}<br>$$<br>期望&amp;方差：<br>$$<br>\begin{align}<br>Expectation: E(X)&amp;=1\cdot p+0\cdot q=p\<br>Variance：Var(X)&amp;2=pq<br>\end{align}<br>$$</p>
<h4 id="2-2-二项分布"><a href="#2-2-二项分布" class="headerlink" title="2.2 二项分布"></a>2.2 二项分布</h4><p>在n次独立重复的<strong>伯努利试验</strong>中，设每次试验中事件A发生的概率为p。用X表示n重伯努利试验中事件A发生的次数，则X的可能取值为0，1，…，n,且对每一个k（0≤k≤n）,事件{X=k}即为“n次试验中事件A恰好发生k次”，随机变量X的离散概率分布即为二项分布。</p>
<p>一般地，如果随机变量服从参数为n和p的二项分布，我们记为$X~B(n,p)$。n次试验中正好得到k次成功的概率由概率质量函数给出：<br>$$<br>P(x=k)=C^k_np^k(1-p)^{n-k}<br>$$<br>期望&amp;方差：<br>$$<br>\begin{align}<br>Expectation: E(X)&amp;=np\<br>Variance：Var(X)&amp;=np(1-p) \<br>\end{align}<br>$$</p>
<h4 id="2-3-泊松分布"><a href="#2-3-泊松分布" class="headerlink" title="2.3 泊松分布"></a>2.3 泊松分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-4-均匀分布"><a href="#2-4-均匀分布" class="headerlink" title="2.4 均匀分布"></a>2.4 均匀分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-5-指数分布"><a href="#2-5-指数分布" class="headerlink" title="2.5 指数分布"></a>2.5 指数分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-6-正态分布"><a href="#2-6-正态分布" class="headerlink" title="2.6 正态分布"></a>2.6 正态分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-7-多项式分布"><a href="#2-7-多项式分布" class="headerlink" title="2.7 多项式分布"></a>2.7 多项式分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-8-alpha分布"><a href="#2-8-alpha分布" class="headerlink" title="2.8 alpha分布"></a>2.8 alpha分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-9-gamma分布"><a href="#2-9-gamma分布" class="headerlink" title="2.9 gamma分布"></a>2.9 gamma分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-10-beta分布"><a href="#2-10-beta分布" class="headerlink" title="2.10 beta分布"></a>2.10 beta分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-11-狄雷克雷分布"><a href="#2-11-狄雷克雷分布" class="headerlink" title="2.11 狄雷克雷分布"></a>2.11 狄雷克雷分布</h4><p>$$</p>
<p>$$</p>
<h4 id="2-11-超几何分布"><a href="#2-11-超几何分布" class="headerlink" title="2.11 超几何分布"></a>2.11 超几何分布</h4><p>$$</p>
<p>$$</p>
<h3 id="联合概率-边缘概率-条件概率"><a href="#联合概率-边缘概率-条件概率" class="headerlink" title="联合概率/边缘概率/条件概率"></a>联合概率/边缘概率/条件概率</h3><h4 id="联合概率"><a href="#联合概率" class="headerlink" title="联合概率"></a>联合概率</h4><figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">联合概率指的是包含多个条件且所有条件同时成立的概率，记作P(X=a,Y=b)或P(a,b)，有的书上也习惯记作P(ab)，但是这种记法个人不太习惯，所以下文采用以逗号分隔的记法。</span><br></pre></td></tr></table></figure>

<h4 id="边缘概率"><a href="#边缘概率" class="headerlink" title="边缘概率"></a>边缘概率</h4><figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">边缘概率是与联合概率对应的，P(X=a)或P(Y=b)，这类仅与单个随机变量有关的概率称为边缘概率</span><br></pre></td></tr></table></figure>

<p><strong>边缘概率与联合概率</strong><br>$$<br>\cases{p(X=a)=\sum_{b}p(X=a,X=b) \<br>p(X=b)=\sum_{a}p(X=a,X=b)}<br>$$</p>
<h4 id="条件概率"><a href="#条件概率" class="headerlink" title="条件概率"></a>条件概率</h4><figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">条件概率表示在条件Y=b成立的情况下，X=a的概率，记作P(X=a|Y=b)或P(a|b)，它具有如下性质：</span><br><span class="line">“在条件Y=b下X的条件分布”也是一种“X的概率分布”，因此穷举X的可取值之后，所有这些值对应的概率之和为1即：</span><br></pre></td></tr></table></figure>

<p>$$<br>∑_aP(X=a|Y=b)=1<br>$$</p>
<p>联合概率/边缘概率/条件概率<br>$$<br>p(X=a|Y=b)=\frac{p(X=a,Y=b)}{p(Y=b)}<br>$$</p>
<h3 id="数理统计"><a href="#数理统计" class="headerlink" title="数理统计"></a>数理统计</h3><h4 id="1-大数定理"><a href="#1-大数定理" class="headerlink" title="1 大数定理"></a>1 大数定理</h4><p>弱大数定律（辛钦大数定律）：<br>设 $X_1,X_2,⋯ $是相互独立，服从同一分布的随机变量序列，且具有数学期望$E(X_k)=\mu(k=1,2,\cdots)$，作前 n 个变量的算术平均$\frac{1}{n}\sum_{k=1}^{n}X_{k}$，则对于任意 ε&gt;0，有<br>$$<br>\lim_{n\rightarrow\infty}P\left{ |\frac{1}{n}\sum_{k=1}^{n}X_{k}-\mu|&lt;\varepsilon\right} =1<br>$$<br>解释：大量实验证实，随机事件A 的频率 $f_n(A)$ 当重复实验次数 n增大时总呈现出稳定性，稳定在某一个常数附近，频率的稳定性是概率定义的可观基础。通俗地说，辛钦大数定律是说，对于独立同分布且具有均值的随机变量 $X_1,X_2,⋯,X_n$, <strong>当 n 很大时它们的算术平均很可能接近于它们的期望值 μ</strong>。</p>
<p>伯努利大数定律（辛钦大数定律的推论）：</p>
<p>设 $f_A$ 是 n 次独立重复实验事件中 A 发生的次数，p是事件 A 在每次试验中发生的概率，则对于任意正数 ε&gt;0，有<br>$$<br>\lim_{n\rightarrow\infty}P\left{ |\frac{f_{A}}{n}-p|&lt;\varepsilon\right} =1<br>$$<br>or<br>$$<br>\lim_{n\rightarrow\infty}P\left{ |\frac{f_{A}}{n}-p|\geq\varepsilon\right} =0<br>$$<br>解释：事件$\left{ |\frac{f_{A}}{n}-p|&lt;\varepsilon\right}$ 是一个小概率事件，这一事件在一次试验中实际上几乎是不发生的，但是当 n 充分大时，该事件几乎是必定发生，也就是说对于给定的任意小的正数 ε, 在 n 充分大时，事件“频率$\frac{f_A}{n}$与概率 p 的偏差小于 ε”实际上几乎必定要发生。在实际应用中，<strong>当实验次数很大时，便可以用事件的频率来替代事件的概率</strong>。</p>
<h4 id="2-中心极限定理"><a href="#2-中心极限定理" class="headerlink" title="2 中心极限定理"></a>2 中心极限定理</h4><p>独立同分布的中心极限定理</p>
<p>设随机变量$X_{1},X_{2},\cdots,X_{n},\cdots$相互独立，服从同一分布，且具有数学期望和方差：</p>
<p>$E(X_{k})=\mu,D(X_{k})=\sigma^{2}&gt;0(k=1,2,\cdots)$，则随机变量之和$\Sigma^2_{k=1}X_k$的标准化变量<br>$$<br>Y_{n}=\frac{\sum_{k=1}^{n}X_{K}-E(\sum_{k=1}^{n}X_{k})}{\sqrt{D(\sum_{k=1}^{n}X_{k})}}=\frac{\sum_{k=1}^{n}X_{k}-n\mu}{\sqrt{n}\sigma}<br>$$<br>近似地服从正态分布N(0,1)N(0,1)。</p>
<p>即,均值为 μ, 方差为$σ^2&gt;0$的独立同分布的随机变量$X_{1},X_{2},\cdots,X_{n},\cdots$的算术平均$\bar{X}=\frac{1}{n}\sum_{k=1}^{n}X_{k}$， <strong>当 n 充分大时近似地服从均值为 μ，方差为 $\frac{\sigma ^2}{n}$的正态分布</strong>。</p>
<p>中心极限定理表明，在相当一般的条件下，当独立随机变量的个数不断增加时，其和的分布趋于正态分布，这一事实阐明了正态分布的重要性，也揭示了为什么在实际应用中会经常遇到正态分布，也就是揭示了产生正态分布变量的源泉. 另一方面，他提供了独立同分布随机变量之和$\sum_{k=1}^{n}X_{k}$（其中$X_k$的方差均存在）的近似分布，<strong>只要和式中加项的个数充分大，可以不必考虑和式中的随机变量服从什么分布，都可以用正态分布来近似</strong>，这在应用上是有效和重要的。</p>

    </div>

    
    
    

      <footer class="post-footer">
          
          <div class="post-tags">
              <a href="/blog/tags/%E6%A6%82%E7%8E%87%E8%AE%BA/" rel="tag"><i class="fa fa-tag"></i> 概率论</a>
          </div>

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/blog/2020/10/09/%E5%9C%A8%E5%85%B3%E7%B3%BB%E4%B8%AD%E8%AE%A4%E8%AF%86%E8%87%AA%E6%88%91/" rel="prev" title="在关系中认识自我（克里希那穆提）">
      <i class="fa fa-chevron-left"></i> 在关系中认识自我（克里希那穆提）
    </a></div>
      <div class="post-nav-item">
    <a href="/blog/2020/11/12/%E4%BD%9C%E5%9B%BE/" rel="next" title="markdown 作流程图">
      markdown 作流程图 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  



          </div>
          
    <div class="comments" id="valine-comments"></div>

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>


        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#1-%E5%9F%BA%E6%9C%AC%E6%A6%82%E7%8E%87%E8%AE%BA%E5%85%AC%E5%BC%8F"><span class="nav-number">1.</span> <span class="nav-text">1. 基本概率论公式</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#1-2-%E6%9C%9F%E6%9C%9B%E3%80%81%E6%96%B9%E5%B7%AE%E3%80%81%E5%8D%8F%E6%96%B9%E5%B7%AE%E3%80%81%E7%9B%B8%E5%85%B3%E7%B3%BB%E6%95%B0"><span class="nav-number">2.</span> <span class="nav-text">1.2 期望、方差、协方差、相关系数</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E6%9C%9F%E6%9C%9B"><span class="nav-number">2.1.</span> <span class="nav-text">期望</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E6%96%B9%E5%B7%AE"><span class="nav-number">2.2.</span> <span class="nav-text">方差</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%8D%8F%E6%96%B9%E5%B7%AE"><span class="nav-number">2.3.</span> <span class="nav-text">协方差</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E7%9B%B8%E5%85%B3%E7%B3%BB%E6%95%B0"><span class="nav-number">2.4.</span> <span class="nav-text">相关系数</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%8D%8F%E6%96%B9%E5%B7%AE%E7%9F%A9%E9%98%B5%E2%80%A6"><span class="nav-number">2.5.</span> <span class="nav-text">协方差矩阵…</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#2-%E6%A6%82%E7%8E%87%E5%88%86%E5%B8%83"><span class="nav-number">3.</span> <span class="nav-text">2. 概率分布</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#2-1-%E4%B8%A4%E7%82%B9%E5%88%86%E5%B8%83-%E4%BC%AF%E5%8A%AA%E5%88%A9%E5%88%86%E5%B8%83"><span class="nav-number">3.1.</span> <span class="nav-text">2.1 两点分布(伯努利分布)</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-2-%E4%BA%8C%E9%A1%B9%E5%88%86%E5%B8%83"><span class="nav-number">3.2.</span> <span class="nav-text">2.2 二项分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-3-%E6%B3%8A%E6%9D%BE%E5%88%86%E5%B8%83"><span class="nav-number">3.3.</span> <span class="nav-text">2.3 泊松分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-4-%E5%9D%87%E5%8C%80%E5%88%86%E5%B8%83"><span class="nav-number">3.4.</span> <span class="nav-text">2.4 均匀分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-5-%E6%8C%87%E6%95%B0%E5%88%86%E5%B8%83"><span class="nav-number">3.5.</span> <span class="nav-text">2.5 指数分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-6-%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83"><span class="nav-number">3.6.</span> <span class="nav-text">2.6 正态分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-7-%E5%A4%9A%E9%A1%B9%E5%BC%8F%E5%88%86%E5%B8%83"><span class="nav-number">3.7.</span> <span class="nav-text">2.7 多项式分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-8-alpha%E5%88%86%E5%B8%83"><span class="nav-number">3.8.</span> <span class="nav-text">2.8 alpha分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-9-gamma%E5%88%86%E5%B8%83"><span class="nav-number">3.9.</span> <span class="nav-text">2.9 gamma分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-10-beta%E5%88%86%E5%B8%83"><span class="nav-number">3.10.</span> <span class="nav-text">2.10 beta分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-11-%E7%8B%84%E9%9B%B7%E5%85%8B%E9%9B%B7%E5%88%86%E5%B8%83"><span class="nav-number">3.11.</span> <span class="nav-text">2.11 狄雷克雷分布</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-11-%E8%B6%85%E5%87%A0%E4%BD%95%E5%88%86%E5%B8%83"><span class="nav-number">3.12.</span> <span class="nav-text">2.11 超几何分布</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E8%81%94%E5%90%88%E6%A6%82%E7%8E%87-%E8%BE%B9%E7%BC%98%E6%A6%82%E7%8E%87-%E6%9D%A1%E4%BB%B6%E6%A6%82%E7%8E%87"><span class="nav-number">4.</span> <span class="nav-text">联合概率&#x2F;边缘概率&#x2F;条件概率</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E8%81%94%E5%90%88%E6%A6%82%E7%8E%87"><span class="nav-number">4.1.</span> <span class="nav-text">联合概率</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E8%BE%B9%E7%BC%98%E6%A6%82%E7%8E%87"><span class="nav-number">4.2.</span> <span class="nav-text">边缘概率</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E6%9D%A1%E4%BB%B6%E6%A6%82%E7%8E%87"><span class="nav-number">4.3.</span> <span class="nav-text">条件概率</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E6%95%B0%E7%90%86%E7%BB%9F%E8%AE%A1"><span class="nav-number">5.</span> <span class="nav-text">数理统计</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#1-%E5%A4%A7%E6%95%B0%E5%AE%9A%E7%90%86"><span class="nav-number">5.1.</span> <span class="nav-text">1 大数定理</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86"><span class="nav-number">5.2.</span> <span class="nav-text">2 中心极限定理</span></a></li></ol></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="bgape002"
      src="/blog/images/head.png">
  <p class="site-author-name" itemprop="name">bgape002</p>
  <div class="site-description" itemprop="description">mail: bgape002@163.com</div>
</div>
<div class="site-state-wrap motion-element">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/blog/archives/">
        
          <span class="site-state-item-count">61</span>
          <span class="site-state-item-name">日志</span>
        </a>
      </div>
      <div class="site-state-item site-state-categories">
            <a href="/blog/categories/">
          
        <span class="site-state-item-count">27</span>
        <span class="site-state-item-name">分类</span></a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/blog/tags/">
          
        <span class="site-state-item-count">66</span>
        <span class="site-state-item-name">标签</span></a>
      </div>
  </nav>
</div>



      </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

        

<div class="copyright">
  
  &copy; 2021 – 
  <span itemprop="copyrightYear">2023</span>
  <span class="with-love">
    <i class="fa fa-heart"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">bgape002</span>
</div>
  <div class="powered-by">由 <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> & <a href="https://theme-next.org/" class="theme-link" rel="noopener" target="_blank">NexT.Gemini</a> 强力驱动
  </div>

        
<div class="busuanzi-count">
  <script async src="https://busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js"></script>
    <span class="post-meta-item" id="busuanzi_container_site_uv" style="display: none;">
      <span class="post-meta-item-icon">
        <i class="fa fa-user"></i>
      </span>
      <span class="site-uv" title="总访客量">
        <span id="busuanzi_value_site_uv"></span>
      </span>
    </span>
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item" id="busuanzi_container_site_pv" style="display: none;">
      <span class="post-meta-item-icon">
        <i class="fa fa-eye"></i>
      </span>
      <span class="site-pv" title="总访问量">
        <span id="busuanzi_value_site_pv"></span>
      </span>
    </span>
</div>








      </div>
    </footer>
  </div>

  
  <script src="/blog/lib/anime.min.js"></script>
  <script src="/blog/lib/velocity/velocity.min.js"></script>
  <script src="/blog/lib/velocity/velocity.ui.min.js"></script>

<script src="/blog/js/utils.js"></script>

<script src="/blog/js/motion.js"></script>


<script src="/blog/js/schemes/pisces.js"></script>


<script src="/blog/js/next-boot.js"></script>




  




  
<script src="/blog/js/local-search.js"></script>











<script>
if (document.querySelectorAll('pre.mermaid').length) {
  NexT.utils.getScript('//cdn.jsdelivr.net/npm/mermaid@8/dist/mermaid.min.js', () => {
    mermaid.initialize({
      theme    : 'forest',
      logLevel : 3,
      flowchart: { curve     : 'linear' },
      gantt    : { axisFormat: '%m/%d/%Y' },
      sequence : { actorMargin: 50 }
    });
  }, window.mermaid);
}
</script>


  

  
      

<script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      loader: {
          load: ['[tex]/mhchem'],
        source: {
          '[tex]/amsCd': '[tex]/amscd',
          '[tex]/AMScd': '[tex]/amscd'
        }
      },
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
          packages: {'[+]': ['mhchem']},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              let target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    (function () {
      var script = document.createElement('script');
      script.src = '//cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js';
      script.defer = true;
      document.head.appendChild(script);
    })();
  } else {
    MathJax.startup.document.state(0);
    MathJax.texReset();
    MathJax.typeset();
  }
</script>

    

  


<script>
NexT.utils.loadComments(document.querySelector('#valine-comments'), () => {
  NexT.utils.getScript('//unpkg.com/valine/dist/Valine.min.js', () => {
    var GUEST = ['nick', 'mail', 'link'];
    var guest = 'nick,mail';
    guest = guest.split(',').filter(item => {
      return GUEST.includes(item);
    });
    new Valine({
      el         : '#valine-comments',
      verify     : false,
      notify     : false,
      appId      : 'szhBf0Qamzsowubi1WnkXmUj-gzGzoHsz',
      appKey     : 'WpwxoK0fVJHvhzWwakr9vbpA',
      placeholder: "Just go go",
      avatar     : 'mm',
      meta       : guest,
      pageSize   : '10' || 10,
      visitor    : false,
      lang       : '' || 'zh-cn',
      path       : location.pathname,
      recordIP   : false,
      serverURLs : ''
    });
  }, window.Valine);
});
</script>

</body>
</html>
